The Stacks project

Proof. The proof mirrors the proof of Lemma 80.12.1. Let $R = U \times _ F U$. By (2) the presheaf $R$ is an algebraic space and by (3) the projections $R \to U$ are smooth and surjective. Denote $(U, R, s, t, c)$ the groupoid associated to the equivalence relation $j : R \to U \times _ S U$ (see Groupoids in Spaces, Lemma 78.11.3). By Theorem 80.10.1 we see that $X = U/R$ (quotient in the fppf-topology) is an algebraic space. Using that the smooth topology and the étale topology have the same sheaves (by More on Morphisms, Lemma 37.38.7) we see the map $U \to F$ identifies $F$ as the quotient of $U$ by $R$ for the smooth topology (details omitted). Thus we have morphisms (transformations of functors)

By Lemma 80.11.6 we see that $U \to X$ is surjective, flat and locally of finite presentation. By Groupoids in Spaces, Lemma 78.19.5 (and the fact that $j$ is a monomorphism) we have $R = U \times _ X U$. By Descent on Spaces, Lemma 74.11.26 we conclude that $U \to X$ is smooth and surjective (as the projections $R \to U$ are smooth and surjective and $\{ U \to X\} $ is an fppf covering). Hence for any scheme $T$ and morphism $T \to X$ the fibre product $T \times _ X U$ is an algebraic space surjective and smooth over $T$. Choose a scheme $V$ and a surjective étale morphism $V \to T \times _ X U$. Then $\{ V \to T\} $ is a smooth covering such that $V \to T \to X$ lifts to a morphism $V \to U$. This proves that $U \to X$ is surjective as a map of sheaves in the smooth topology. It follows that $F \to X$ is surjective as a map of sheaves in the smooth topology. On the other hand, the map $F \to X$ is injective (as a map of presheaves) since $R = U \times _ X U$. It follows that $F \to X$ is an isomorphism of smooth ($=$ étale) sheaves, see Sites, Lemma 7.11.2 which concludes the proof. $\square$